, the approximation is based on expansions proposed by Cody and Thatcher Jr. (1969). Precautions are taken to maintain good relative accuracy in the vicinity of

x_{0} \approx - 0.372507 \dots

, which corresponds to a simple zero of Ei(

- x

nag_specfun_integral_exp (s13aa) guards against producing underflows and overflows by using the argument

x_{hi}

. To guard against overflow, if

x < - x_{hi}

the function terminates and returns the negative of the largest representable machine number. To guard against underflow, if

x > x_{hi}

the result is set directly to zero.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Cody W J and Thatcher Jr. H C (1969) Rational Chebyshev approximations for the exponential integral Ei

(x)

Math. Comp. 23 289–303

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: The argument $x$ of the function.

Constraint: $- x_{hi} \leq x < 0.0$ or $x > 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: On entry, $x = 0.0$ and the function is infinite. The result returned is the largest representable machine number.

$ifail = 2$: The evaluation has been abandoned due to the likelihood of overflow. The argument $x < - x_{hi}$ , and the result is returned as the negative of the largest representable machine number.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Unless stated otherwise, it is assumed that

x > 0

δ

and

ε

are the relative errors in argument and result respectively, then in principle,

|ε| ≃ |\frac{e^{- x}}{E_{1} (x)} \times δ|

so the relative error in the argument is amplified in the result by at least a factor

e^{- x} / E_{1} (x)

. The equality should hold if

δ

is greater than the machine precision (

δ

due to data errors etc.) but if

δ

is simply a result of round-off in the machine representation, it is possible that an extra figure may be lost in internal calculation and round-off.

The behaviour of this amplification factor is shown in the following graph:

Figure 1

It should be noted that, for absolutely small

x

, the amplification factor tends to zero and eventually the error in the result will be limited by machine precision.

For absolutely large

x

ε \sim x δ = Δ,

the absolute error in the argument.

For

x < 0

, empirical tests have shown that the maximum relative error is a loss of approximately

1

decimal place.

Further Comments

None.

Example

The following program reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

Open in the MATLAB editor: s13aa_example

function s13aa_example


fprintf('s13aa example results\n\n');

x = [2  -1];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s13aa(x(j));
end

disp('      x          E_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s13aa_plot;



function s13aa_plot
  x = [-5:0.1:-0.2, -0.1:0.01:-0.01, -1e-3, -1e-4, -1e-5...
       1e-5, 1e-4, 1e-3, 0.01:0.01:0.1, 0.2:0.1:4.8];
  for j=1:numel(x)
    [e1(j), ifail] = s13aa(x(j));
  end

  fig1 = figure;
  plot(x,e1);
  xlabel('x');
  ylabel('E_1(x)');
  title('Exponential Integral E_1(x)');

s13aa example results

      x          E_1(x)
   2.000e+00   4.890e-02
  -1.000e+00  -1.895e+00

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox